DIFFERENTIAL EQUATIONS FOR ENGINEERS

This book presents a systematic and comprehensive introduction to ordinary differentialequations for engineering students and practitioners. Mathematical concepts and varioustechniques are presented in a clear, logical, and concise manner. Various visual featuresare used to highlight focus areas. Complete illustrative diagrams are used to facilitatemathematical modeling of application problems. Readers are motivated by a focus onthe relevance of differential equations through their applications in various engineeringdisciplines. Studies of various types of differential equations are determined by engi-neering applications. Theory and techniques for solving differential equations are thenapplied to solve practical engineering problems. Detailed step-by-step analysis is pre-sented to model the engineering problems using differential equations from physicalprinciples and to solve the differential equations using the easiest possible method. Sucha detailed, step-by-step approach, especially when applied to practical engineering prob-lems, helps the readers to develop problem-solving skills.

This book is suitable for use not only as a textbook on ordinary differential equa-tions for undergraduate students in an engineering program but also as a guide to self-study. It can also be used as a reference after students have completed learning thesubject.

Wei-Chau Xie is a Professor in the Department of Civil and Environment Engineeringand the Department of Applied Mathematics at the University of Waterloo. He is theauthor of Dynamic Stability of Structures and has published numerous journal articleson dynamic stability, structural dynamics and random vibration, nonlinear dynamics andstochastic mechanics, reliability and safety analysis of engineering systems, and seismicanalysis and design of engineering structures. He has been teaching differential equa-tions to engineering students for almost twenty years. He received the Teaching Excel-lence Award in 2001 in recognition of his exemplary record of outstanding teaching,concern for students, and commitment to the development and enrichment of engineer-ing education at Waterloo. He is the recipient of the Distinguished Teacher Award in2007, which is the highest formal recognition given by the University of Waterloo for asuperior record of continued excellence in teaching.

Differential Equations for Engineers

Wei-Chau XieUniversity of Waterloo

cambridge university pressCambridge, New York, Melbourne, Madrid, Cape Town, Singapore,Sao Paulo, Delhi, Dubai, Tokyo

Cambridge University Press32 Avenue of the Americas, New York, NY 10013-2473, USA

www.cambridge.orgInformation on this title: www.cambridge.org/9780521194242

© Wei-Chau Xie 2010

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.

First published 2010

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication data

Xie, Wei-Chau, 1964–Differential equations for engineers / Wei-Chau Xie.

p. cm.Includes bibliographical references and index.ISBN 978-0-521-19424-21. Differential equations. 2. Engineering mathematics. I. Title.TA347.D45X54 2010620.001′515352–dc22 2010001101

ISBN 978-0-521-19424-2 Hardback

Cambridge University Press has no responsibility for thepersistence or accuracy of URLs for external or third-party InternetWeb sites referred to in this publication and does not guarantee thatany content on such Web sites is, or will remain, accurate orappropriate.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivating Examples 1

1.2 General Concepts and Definitions 6

2 First-Order and Simple Higher-Order Differential Equations . 16

2.1 The Method of Separation of Variables 16

2.2 Method of Transformation of Variables 20

2.2.1 Homogeneous Equations 20

2.2.2 Special Transformations 25

2.3 Exact Differential Equations and Integrating Factors 31

2.3.1 Exact Differential Equations 32

2.3.2 Integrating Factors 39

2.3.3 Method of Inspection 45

2.3.4 Integrating Factors by Groups 48

2.4 Linear First-Order Equations 55

2.4.1 Linear First-Order Equations 55

2.4.2 Bernoulli Differential Equations 58

2.5 Equations Solvable for the Independent or Dependent Variable 61

2.6 Simple Higher-Order Differential Equations 68

2.6.1 Equations Immediately Integrable 68

2.6.2 The Dependent Variable Absent 70

2.6.3 The Independent Variable Absent 72

2.7 Summary 74

Problems 78

3 Applications of First-Order and Simple Higher-Order Equations 87

3.1 Heating and Cooling 87

3.2 Motion of a Particle in a Resisting Medium 91

3.3 Hanging Cables 97

vii

viii contents

3.3.1 The Suspension Bridge 97

3.3.2 Cable under Self-Weight 102

3.4 Electric Circuits 108

3.5 Natural Purification in a Stream 114

3.6 Various Application Problems 120

Problems 130

4 Linear Differential Equations . . . . . . . . . . . . . . . . . . . 140

4.1 General Linear Ordinary Differential Equations 140

4.2 Complementary Solutions 143

4.2.1 Characteristic Equation Having Real Distinct Roots 143

4.2.2 Characteristic Equation Having Complex Roots 147

4.2.3 Characteristic Equation Having Repeated Roots 151

4.3 Particular Solutions 153

4.3.1 Method of Undetermined Coefficients 153

4.3.2 Method of Operators 162

4.3.3 Method of Variation of Parameters 173

4.4 Euler Differential Equations 178

4.5 Summary 180

Problems 183

5 Applications of Linear Differential Equations . . . . . . . . . . 188

5.1 Vibration of a Single Degree-of-Freedom System 188

5.1.1 Formulation—Equation of Motion 188

5.1.2 Response of a Single Degree-of-Freedom System 193

5.1.2.1 Free Vibration—Complementary Solution 193

5.1.2.2 Forced Vibration—Particular Solution 200

5.2 Electric Circuits 209

5.3 Vibration of a Vehicle Passing a Speed Bump 213

5.4 Beam-Columns 218

5.5 Various Application Problems 223

Problems 232

6 The Laplace Transform and Its Applications . . . . . . . . . . . 244

6.1 The Laplace Transform 244

contents ix

6.2 The Heaviside Step Function 249

6.3 Impulse Functions and the Dirac Delta Function 254

6.4 The Inverse Laplace Transform 257

6.5 Solving Differential Equations Using the Laplace Transform 263

6.6 Applications of the Laplace Transform 268

6.6.1 Response of a Single Degree-of-Freedom System 268

6.6.2 Other Applications 275

6.6.3 Beams on Elastic Foundation 283

6.7 Summary 289

Problems 291

7 Systems of Linear Differential Equations . . . . . . . . . . . . 300

7.1 Introduction 300

7.2 The Method of Operator 304

7.2.1 Complementary Solutions 304

7.2.2 Particular Solutions 307

7.3 The Method of Laplace Transform 318

7.4 The Matrix Method 325

7.4.1 Complementary Solutions 326

7.4.2 Particular Solutions 334

7.4.3 Response of Multiple Degrees-of-Freedom Systems 344

7.5 Summary 347

7.5.1 The Method of Operator 347

7.5.2 The Method of Laplace Transform 348

7.5.3 The Matrix Method 349

Problems 351

8 Applications of Systems of Linear Differential Equations . . . 357

8.1 Mathematical Modeling of Mechanical Vibrations 357

8.2 Vibration Absorbers or Tuned Mass Dampers 366

8.3 An Electric Circuit 372

8.4 Vibration of a Two-Story Shear Building 377

8.4.1 Free Vibration—Complementary Solutions 378

8.4.2 Forced Vibration—General Solutions 380

Problems 384

x contents

9 Series Solutions of Differential Equations . . . . . . . . . . . . 390

9.1 Review of Power Series 391

9.2 Series Solution about an Ordinary Point 394

9.3 Series Solution about a Regular Singular Point 403

9.3.1 Bessel’s Equation and Its Applications 408

9.3.1.1 Solutions of Bessel’s Equation 408

9.3.2 Applications of Bessel’s Equation 418

9.4 Summary 424

Problems 426

10 Numerical Solutions of Differential Equations . . . . . . . . . 431

10.1 Numerical Solutions of First-Order Initial Value Problems 431

10.1.1 The Euler Method or Constant Slope Method 432

10.1.2 Error Analysis 434

10.1.3 The Backward Euler Method 436

10.1.4 Improved Euler Method—Average Slope Method 437

10.1.5 The Runge-Kutta Methods 440

10.2 Numerical Solutions of Systems of Differential Equations 445

10.3 Stiff Differential Equations 449

10.4 Summary 452

Problems 454

11 Partial Differential Equations . . . . . . . . . . . . . . . . . . . 457

11.1 Simple Partial Differential Equations 457

11.2 Method of Separation of Variables 458

11.3 Application—Flexural Motion of Beams 465

11.3.1 Formulation—Equation of Motion 465

11.3.2 Free Vibration 466

11.3.3 Forced Vibration 471

11.4 Application—Heat Conduction 473

11.4.1 Formulation—Heat Equation 473

11.4.2 Two-Dimensional Steady-State Heat Conduction 476

11.4.3 One-Dimensional Transient Heat Conduction 480

11.4.4 One-Dimensional Transient Heat Conduction on a Semi-Infinite

Interval 483

contents xi

11.4.5 Three-Dimensional Steady-State Heat Conduction 488

11.5 Summary 492

Problems 493

12 Solving Ordinary Differential Equations Using Maple . . . . . 498

12.1 Closed-Form Solutions of Differential Equations 499

12.1.1 Simple Ordinary Differential Equations 499

12.1.2 Linear Ordinary Differential Equations 506

12.1.3 The Laplace Transform 507

12.1.4 Systems of Ordinary Differential Equations 509

12.2 Series Solutions of Differential Equations 512

12.3 Numerical Solutions of Differential Equations 517

Problems 526

Appendix A Tables of Mathematical Formulas . . . . . . . . . . . . . 531

a.1 Table of Trigonometric Identities 531

a.2 Table of Derivatives 533

a.3 Table of Integrals 534

a.4 Table of Laplace Transforms 537

a.5 Table of Inverse Laplace Transforms 539

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

Preface

Background

Differential equations have wide applications in various engineering and sciencedisciplines. In general, modeling of the variation of a physical quantity, such astemperature, pressure, displacement, velocity, stress, strain, current, voltage, orconcentration of a pollutant, with the change of time or location, or both wouldresult in differential equations. Similarly, studying the variation of some physicalquantities on other physical quantities would also lead to differential equations.In fact, many engineering subjects, such as mechanical vibration or structuraldynamics, heat transfer, or theory of electric circuits, are founded on the theory ofdifferential equations. It is practically important for engineers to be able to modelphysical problems using mathematical equations, and then solve these equations sothat the behavior of the systems concerned can be studied.

I have been teaching differential equations to engineering students for the pasttwo decades. Most, if not all, of the textbooks are written by mathematicianswith little engineering background. Based on my experience and feedback fromstudents, the following lists some of the gaps frequently seen in current textbooks:

❧ A major focus is put on explaining mathematical concepts

For engineers, the purpose of learning the theory of differential equations isto be able to solve practical problems where differential equations are used.For engineering students, it is more important to know the applications andtechniques for solving application problems than to delve into the nuances ofmathematical concepts and theorems. Knowing the appropriate applications canmotivate them to study the mathematical concepts and techniques. However,it is much more challenging to model an application problem using physicalprinciples and then solve the resulting differential equations than it is to merelycarry out mathematical exercises.

❧ Insufficient emphasis is placed on the step-by-step problem solving techniques

Engineering students do not usually have the same mathematical backgroundand interest as students who major in mathematics. Mathematicians are moreinterested if: (1) there are solutions to a differential equation or a system ofdifferential equations; (2) the solutions are unique under a certain set of con-ditions; and (3) the differential equations can be solved. On the other hand,

xiii

xiv preface

engineers are more interested in mathematical modeling of a practical problemand actually solving the equations to find the solutions using the easiest possiblemethod. Hence, a detailed step-by-step approach, especially applied to practicalengineering problems, helps students to develop problem solving skills.

❧ Presentations are usually formula-driven with little variation in visual design

It is very difficult to attract students to read boring formulas without variationof presentation. Readers often miss the points of importance.

Objectives

This book addresses the needs of engineering students and aims to achieve thefollowing objectives:

❧ To motivate students on the relevance of differential equations in engineeringthrough their applications in various engineering disciplines. Studies of varioustypes of differential equations are motivated by engineering applications; the-ory and techniques for solving differential equations are then applied to solvepractical engineering problems.

❧ To have a balance between theory and applications. This book could be used as areference after students have completed learning the subject. As a reference, it hasto be reasonably comprehensive and complete. Detailed step-by-step analysis ispresented to model the engineering problems using differential equations andto solve the differential equations.

❧ To present the mathematical concepts and various techniques in a clear, logicaland concise manner. Various visual features, such as side-notes (preceded bythe symbol), different fonts and shades, are used to highlight focus areas.Complete illustrative diagrams are used to facilitate mathematical modeling ofapplication problems. This book is not only suitable as a textbook for classroomuse but also is easy for self-study. As a textbook, it has to be easy to understand.For self-study, the presentation is detailed with all necessary steps and usefulformulas given as side-notes.

Scope

This book is primarily for engineering students and practitioners as the mainaudience. It is suitable as a textbook on ordinary differential equations for under-graduate students in an engineering program. Such a course is usually offered inthe second year after students have taken calculus and linear algebra in the firstyear. Although it is assumed that students have a working knowledge of calculusand linear algebra, some important concepts and results are reviewed when they arefirst used so as to refresh their memory.

preface xv

Chapter 1 first presents some motivating examples, which will be studied indetail later in the book, to illustrate how differential equations arise in engineer-ing applications. Some basic general concepts of differential equations are thenintroduced.

In Chapter 2, various techniques for solving first-order and simple higher-orderordinary differential equations are presented. These methods are then applied inChapter 3 to study various application problems involving first-order and simplehigher-order differential equations.

Chapter 4 studies linear ordinary differential equations. Complementary solu-tions are obtained through the characteristic equations and characteristic numbers.Particular solutions are obtained using the method of undetermined coefficients,the operator method, and the method of variation of parameters. Applicationsinvolving linear ordinary differential equations are presented in Chapter 5.

Solutions of linear ordinary differential equations using the Laplace transformare studied in Chapter 6, emphasizing functions involving Heaviside step functionand Dirac delta function.

Chapter 7 studies solutions of systems of linear ordinary differential equations.The method of operator, the method of Laplace transform, and the matrix methodare introduced. Applications involving systems of linear ordinary differential equa-tions are considered in Chapter 8.

In Chapter 9, solutions of ordinary differential equations in series about anordinary point and a regular singular point are presented. Applications of Bessel’sequation in engineering are considered.

Some classical methods, including forward and backward Euler method, im-proved Euler method, and Runge-Kutta methods, are presented in Chapter 10 fornumerical solutions of ordinary differential equations.

In Chapter 11, the method of separation of variables is applied to solve partialdifferential equations. When the method is applicable, it converts a partial differ-ential equation into a set of ordinary differential equations. Flexural vibration ofbeams and heat conduction are studied as examples of application.

Solutions of ordinary differential equations using Maple are presented in Chapter12. Symbolic computation software, such as Maple, is very efficient in solvingproblems involving ordinary differential equations. However, it cannot replacelearning and thinking, especially mathematical modeling. It is important to developanalytical skills and proficiency through “hand” calculations, as has been done inprevious chapters. This will also help the development of insight into the problemsand appreciation of the solution process. For this reason, solutions of ordinarydifferential equations using Maple is presented in the last chapter of the bookinstead of a scattering throughout the book.

xvi preface

The book covers a wide range of materials on ordinary differential equationsand their engineering applications. There are more than enough materials for aone-term (semester) undergraduate course. Instructors can select the materialsaccording to the curriculum. Drafts of this book were used as the textbook in aone-term undergraduate course at the University of Waterloo.

Acknowledgments

First and foremost, my sincere appreciation goes to my students. It is the studentswho give me a stage where I can cultivate my talent and passion for teaching. It isfor the students that this book is written, as my small contribution to their successin academic and professional careers. My undergraduate students who have usedthe draft of this book as a textbook have made many encouraging comments andconstructive suggestions.

I am very grateful to many people who have reviewed and commented on thebook, including Professor Hong-Jian Lai of West Virginia University, Professors S.T.Ariaratnam, Xin-Zhi Liu, Stanislav Potapenko, and Edward Vrscay of the Universityof Waterloo.

My graduate students Mohamad Alwan, Qinghua Huang, Jun Liu, Shunhao Ni,and Richard Wiebe have carefully read the book and made many helpful and criticalsuggestions.

My sincere appreciation goes to Mr. Peter Gordon, Senior Editor, Engineering,Cambridge University Press, for his encouragement, trust, and hard work to publishthis book.

Special thanks are due to Mr. John Bennett, my mentor, teacher, and friend, forhis advice and guidance. He has also painstakingly proofread and copyedited thisbook.

Without the unfailing love and support of my mother, who has always believed inme, this work would not have been possible. In addition, the care, love, patience, andunderstanding of my wife Cong-Rong and lovely daughters Victoria and Tiffanyhave been of inestimable encouragement and help. I love them very much andappreciate all that they have contributed to my work.

I appreciate hearing your comments through email (xie@uwaterloo.ca) or regu-lar correspondence.

Wei-Chau Xie

Waterloo, Ontario, Canada

1C H A P T E R

Introduction

1.1 Motivating Examples

Differential equations have wide applications in various engineering and sciencedisciplines. In general, modeling variations of a physical quantity, such as tempera-ture, pressure, displacement, velocity, stress, strain, or concentration of a pollutant,with the change of time t or location, such as the coordinates (x, y, z), or bothwould require differential equations. Similarly, studying the variation of a physi-cal quantity on other physical quantities would lead to differential equations. Forexample, the change of strain on stress for some viscoelastic materials follows adifferential equation.

It is important for engineers to be able to model physical problems using mathe-matical equations, and then solve these equations so that the behavior of the systemsconcerned can be studied.

In this section, a few examples are presented to illustrate how practical problemsare modeled mathematically and how differential equations arise in them.

Motivating Example 1

First consider the projectile of a mass m launched with initial velocity v0 at angleθ0 at time t = 0, as shown.

O A

y

θ0 x

v0 θθ x

v(t)v(t)

βv mg

y

1

2 1 introduction

The atmosphere exerts a resistance force on the mass, which is proportionalto the instantaneous velocity of the mass, i.e., R =βv, where β is a constant,and is opposite to the direction of the velocity of the mass. Set up the Cartesiancoordinate system as shown by placing the origin at the point from where the massm is launched.

At time t, the mass is at location(x(t), y(t)

). The instantaneous velocity of the

mass in the x- and y-directions are x(t) and y(t), respectively. Hence the velocityof the mass is v(t)=√

x2(t)+ y2(t) at the angle θ(t)= tan−1[

y(t)/x(t)].

The mass is subjected to two forces: the vertical downward gravity mg and theresistance force R(t)=βv(t).

The equations of motion of the mass can be established using Newton’s SecondLaw: F =∑

ma. The x-component of the resistance force is −R(t) cos θ(t). Inthe y-direction, the component of the resistance force is −R(t) sin θ(t). Hence,applying Newton’s Second Law yields

x-direction: max=∑

Fx =⇒ mx(t) = −R(t) cos θ(t),

y-direction: may = ∑Fy =⇒ m y(t) = −mg − R(t) sin θ(t).

Since

θ(t) = tan−1 y(t)

x(t)=⇒ cos θ = x(t)√

x2(t)+ y2(t), sin θ = y(t)√

x2(t)+ y2(t),

the equations of motion become

mx(t) = −βv(t) · x(t)√x2(t)+ y2(t)

=⇒ mx(t)+ β x(t) = 0,

m y(t) = −mg − βv(t) · y(t)√x2(t)+ y2(t)

=⇒ m y(t)+ β y(t) = −mg ,

in which the initial conditions are at time t = 0: x(0)= 0, y(0)= 0, x(0)= v0 cos θ0,y(0)= v0 sin θ0. The equations of motion are two equations involving the first- andsecond-order derivatives x(t), y(t), x(t), and y(t). These equations are called, aswill be defined later, a system of two second-order ordinary differential equations.

Because of the complexity of the problems, in the following examples, the prob-lems are described and the governing equations are presented without detailedderivation. These problems will be investigated in details in later chapters whenapplications of various types of differential equations are studied.

Motivating Example 2

A tank contains a liquid of volume V(t), which is polluted with a pollutant concen-tration in percentage of c(t) at time t. To reduce the pollutant concentration, an

1.1 motivating examples 3

inflow of rate Qin is injected to the tank. Unfortunately, the inflow is also pollutedbut to a lesser degree with a pollutant concentration cin. It is assumed that theinflow is perfectly mixed with the liquid in the tank instantaneously. An outflowof rate Qout is removed from the tank as shown. Suppose that, at time t = 0, thevolume of the liquid is V0 with a pollutant concentration of c0.

Inflow

OutflowVolume V(t)

Concentration c(t)Qout, c(t)

Qin, cin

The equation governing the pollutant concentration c(t) is given by

[V0 + (Qin −Qout)t

] dc(t)

dt+ Qinc(t) = Qincin,

with initial condition c(0)= c0. This is a first-order ordinary differential equation.

Motivating Example 3

Hanger

Deck

Cable

w(x)

O

y

x

Consider the suspension bridge as shown, which consists of the main cable, thehangers, and the deck. The self-weight of the deck and the loads applied on thedeck are transferred to the cable through the hangers.

4 1 introduction

Set up the Cartesian coordinate system by placing the origin O at the lowest pointof the cable. The cable can be modeled as subjected to a distributed load w(x). Theequation governing the shape of the cable is given by

d2y

dx2 = w(x)

H,

where H is the tension in the cable at the lowest point O. This is a second-orderordinary differential equation.

Motivating Example 4

k

Reference position m

c

x(t)

x0(t) y(t)

Consider the vibration of a single-story shear building under the excitation ofearthquake. The shear building consists of a rigid girder of mass m supported bycolumns of combined stiffness k. The vibration of the girder can be described bythe horizontal displacement x(t). The earthquake is modeled by the displacementof the ground x0(t) as shown. When the girder vibrates, there is a damping forcedue to the internal friction between various components of the building, given byc[x(t)− x0(t)

], where c is the damping coefficient.

The relative displacement y(t)= x(t)−x0(t) between the girder and the groundis governed by the equation

my(t)+ c y(t)+ k y(t) = −mx0(t),

which is a second-order linear ordinary differential equation.

Motivating Example 5

In many engineering applications, an equipment of mass m is usually mounted ona supporting structure that can be modeled as a spring of stiffness k and a damperof damping coefficient c as shown in the following figure. Due to unbalanced massin rotating components or other excitation mechanisms, the equipment is subjectedto a harmonic force F0 sin�t. The vibration of the mass is described by the verticaldisplacement x(t). When the excitation frequency � is close toω0 =√

k/m, whichis the natural circular frequency of the equipment and its support, vibration of largeamplitudes occurs.

1.1 motivating examples 5

In order to reduce the vibration of the equipment, a vibration absorber ismounted on the equipment. The vibration absorber can be modeled as a massma, a spring of stiffness ka, and a damper of damping coefficient ca. The vibrationof the absorber is described by the vertical displacement xa(t).

x(t)

Vibration

Absorber

Supporting

Structure

Equipment

xa(t)

F0 sin�t

c

m

k

ca

ma

ka

The equations of motion governing the vibration of the equipment and theabsorber are given by

mx + (c +ca) x + (k+ka)x − caxa − kaxa = F0 sin�t,

maxa + caxa + kaxa − cax − kax = 0,

which comprises a system of two coupled second-order linear ordinary differentialequations.

Motivating Example 6

L

v

PP

EI, ρA

Utt=0

x

A bridge may be modeled as a simply supported beam of length L, mass densityper unit length ρA, and flexural rigidity EI as shown. A vehicle of weight P crossesthe bridge at a constant speed U . Suppose at time t = 0, the vehicle is at the left endof the bridge and the bridge is at rest. The deflection of the bridge is v(x, t), whichis a function of both location x and time t. The equation governing v(x, t) is thepartial differential equation

ρA∂2v(x, t)

∂t2 + EI∂4v(x, t)

∂x4 = P δ(x−U t),

42 2 first-order and simple higher-order differential equations

Example 2.19 2.19

Solve y (cos3x + y sin x)dx + cos x (sin x cos x + 2 y)dy = 0.

The differential equation is of the standard form M dx+N dy = 0, where

M(x, y) = y cos3x + y2 sin x, N(x, y) = sin x cos2x + 2 y cos x.

Test for exactness:

∂M

∂y= cos3x + 2 y sin x,

∂N

∂x= cos3x − 2 sin2x cos x − 2 y sin x,

∴ ∂M

∂y�= ∂N

∂x=⇒ The differential equation is not exact.

Since

1

N

(∂M

∂y− ∂N

∂x

)= (cos3x + 2 y sin x)− (cos3x − 2 sin2x cos x − 2 y sin x)

sin x cos2x + 2 y cos x

= 2 sin x (2 y + sin x cos x)

cos x (2 y + sin x cos x)= 2 sin x

cos x, A function of x only

∴ μ(x) = exp

[∫1

N

(∂M

∂y− ∂N

∂x

)dx

]= exp

[ ∫ 2 sin x

cos xdx

]

= exp[−2

∫1

cos xd(cos x)

]= exp

[−2 ln∣∣cos x

∣∣] = 1

cos2x.

Multiplying the differential equation by the integrating factor μ(x)= 1

cos2xyields

(y cos x + y2 sin x

cos2x

)dx +

(sin x + 2 y

cos x

)dy = 0.

The general solution is determined using the method of grouping terms

(y cos x dx

∫dx

��

+ sin x dy)

y sin x ∂∂y

��+

( 2 y

cos xdy

∫dy ��

+ y2 sin x

cos2xdx

)

y2

cos x

∂∂x

��

= 0,

which gives

y sin x + y2

cos x= C. General solution

120 3 applications of first-order and simple higher-order equations

Procedure for Solving an Application Problem

1. Establish the governing differential equations based on physical principles

and geometrical properties underlying the problem.

2. Identify the type of these differential equations and then solve them.

3. Determine the arbitrary constants in the general solutions using the initial

or boundary conditions.

3.6 Various Application Problems

Example 3.10 — Ferry Boat 3.10

A ferry boat is crossing a river of width a from point A to point O as shown in thefollowing figure. The boat is always aiming toward the destination O. The speed ofthe river flow is constant vR and the speed of the boat is constant vB. Determinethe equation of the path traced by the boat.

vR

vB

vB cosθ

θ

vB sinθ

xx

yRiver Flow

y

P(x,y)

AH aO

Suppose that, at time t, the boat is at point P with coordinates (x, y). The velocityof the boat has two components: the velocity of the boat vB relative to the river flow(as if the river is not flowing), which is pointing toward the origin O or along linePO, and the velocity of the river vR in the y direction.

Decompose the velocity components vB and vR in the x- and y-directions

vx = −vB cos θ , vy = vR − vB sin θ.

From �OHP, it is easy to see

cos θ = OH

OP= x√

x2 +y2, sin θ = PH

OP= y√

x2 +y2.

Hence, the equations of motion are given by

vx = dx

dt= −vB

x√x2 +y2

, vy = dy

dt= vR − vB

y√x2 +y2

.

122 3 applications of first-order and simple higher-order equations

Example 3.11 — Bar with Variable Cross-Section 3.11

A bar with circular cross-sections is supported at the top end and is subjected to aload of P as shown in Figure 3.14(a). The length of the bar is L. The weight densityof the materials is ρ per unit volume. It is required that the stress at every point isconstant σa. Determine the equation for the cross-section of the bar.

x

y

P P

L

x

x

dx

y

y

P+W(x)

(a) (b) (c)

x

x

y

Figure 3.14 A bar under axial load.

Consider a cross-section at level x as shown in Figure 3.14(b). The correspondingradius is y. The volume of a circular disk of thickness dx is dV =πy2 dx. Thevolume of the segment of bar between 0 and x is

V(x) =∫ x

0πy2dx,

and the weight of this segment is

W(x) = ρV(x) = ρ

∫ x

0πy2dx.

The load applied on cross-section at level x is equal to the sum of the externallyapplied load P and the weight of the segment between 0 and x, i.e.,

F(x) = W(x)+ P = ρ

∫ x

0πy2dx + P.

The normal stress is

σ(x) = F(x)

A(x)= 1

πy2

(ρ

∫ x

0πy2dx + P

)= σa =⇒ ρ

∫ x

0πy2dx + P = σaπy2.

Differentiating with respect to x yields

ρπy2 = σaπ ·2 ydy

dx. Variable separable

172 4 linear differential equations

Example 4.25 4.25

Evaluate yP = 1

(D −2)3 e2x.

Use Theorem 4: φ(D) = (D −2)3, φ(2) = 0,

φ′(D) = 3(D −2)2, φ′(2) = 0,

φ′′(D) = 6(D −2), φ′′(2) = 0,

φ′′′(D) = 6, φ′′′(2) = 6 �= 0.

∴ yP = 1

φ′′′(2)x3 e2x = 1

6 x3 e2x.

Example 4.26 4.26

Solve (D2 + 4 D + 13)y = e−2x sin 3x.

The characteristic equation is λ2 +4λ+13 = 0, which gives

λ = −4 ± √42 − 4×13

2= −2 ± i 3.

Hence the complementary solution is yC = e−2x(A cos 3x + B sin 3x).

Remarks: Note that the right-hand side of the differential equation is containedin the complementary solution. Using the method of undetermined coefficient,the assumed form of a particular solution is x ·e−2x (a cos 3x+b sin 3x).

A particular solution is given by

yP = 1

D2 +4D +13

(e−2x sin 3x

) = e−2x 1

(D −2)2 +4(D −2)+13sin 3x

Theorem 2: take e−2x out of the operator, shift D by −2.

= e−2x 1

D2 +9sin 3x = e−2x Im

[ 1

D2 +9ei3x

].

This can be evaluated using Theorem 4:

φ(D) = D2 +9, φ(i 3) = (i 3)2 +9 = 0,

φ′(D) = 2 D, φ′(i 3) = 2(i 3) = i 6 �= 0.

Hence,

yP = e−2x Im[ 1

φ′(i 3)x ei3x

]Theorem 4

= e−2x Im[ 1

i 6x (cos 3x + i sin 3x)

]= e−2x Im

[− i

6x (cos 3x + i sin 3x)

]= − 1

6 x e−2x cos 3x.

5.2 electric circuits 209

5.2 Electric Circuits

Series RLC Circuit

A circuit consisting of a resistor R, an inductor L, a capacitor C, and a voltagesource V(t) connected in series, shown in Figure 5.18, is called the series RLCcircuit. Applying Kirchhoff ’s Voltage Law, one has

−V(t)+ Ri + Ldi

dt+ 1

C

∫ t

−∞i dt = 0.

V(t)

R

C

L

i

m=L

x(t)= i(t)

F(t)=

k= 1C

c=R

dV(t)dt

Figure 5.18 Series RLC circuit.

Differentiating with respect to t yields

Ld2i

dt2 + Rdi

dt+ 1

Ci = dV(t)

dt,

or, in the standard form,

d2i

dt2 + 2ζω0di

dt+ ω2

0 i = 1

L

dV(t)

dt, ω2

0 = 1

LC, ζω0 = R

2L.

The series RLC circuit is equivalent to a mass-damper-spring system as shown.

Parallel RLC Circuit

A circuit consisting of a resistor R, an inductor L, a capacitor C, and a currentsource I(t) connected in parallel, as shown in Figure 5.19, is called the parallelRLC circuit. Applying Kirchhoff ’s Current Law at node 1, one has

I(t) = Cdv

dt+ 1

L

∫ t

−∞v dt + v

R.

Differentiating with respect to t yields

Cd2v

dt2 + 1

R

dv

dt+ 1

Lv = dI(t)

dt,

or, in the standard form,

d2v

dt2 + 2ζω0dv

dt+ ω2

0 v = 1

C

dI(t)

dt, ω2

0 = 1

LC, ζω0 = 1

2RC.

The parallel RLC circuit is equivalent to a mass-damper-spring system as shown.

210 5 applications of linear differential equations

R L

I(t)

v

1

m=C

x(t)=v(t)

F(t)=

k= 1L

dI(t)dtC c= 1

R

Figure 5.19 Parallel RLC circuit.

Example 5.1 — Automobile Ignition Circuit 5.1

An automobile ignition system is modeled by the circuit shown in the followingfigure. The voltage source V0 represents the battery and alternator. The resistor Rmodels the resistance of the wiring, and the ignition coil is modeled by the inductorL. The capacitor C, known as the condenser, is in parallel with the switch, which isknown as the electronic ignition. The switch has been closed for a long time priorto t<0−. Determine the inductor voltage vL for t>0.

V0

t=0

R C

L

Spark PlugIgnition Coil

vCi

vL

For V0 = 12 V, R = 4�, C = 1μF, L = 8 mH, determine the maximal inductorvoltage and the time when it is reached.

❧ For t<0, the switch is closed, the capacitor behaves as an open circuit and theinductor behaves as a short circuit as shown. Hence i(0−)= V0/R, vC(0

−)= 0.

V0

R

vC(0−) i(0−)

vL(0−)

t 0– t 0+

V0

R C

L

Ignition Coil

Mesh

vCi

vL

❧ At t = 0, the switch is opened. Since the current in an inductor and the voltageacross a capacitor cannot change abruptly, one has i(0+)= i(0−)= V0/R, vC(0

+)=vC(0

−)= 0. The derivative i ′(0+) is obtained from vL(0+), which is determined by

5.5 various application problems 223

5.5 Various Application Problems

Example 5.5 — Jet Engine Vibration 5.5

As shown in Figure 5.8, jet engines are supported by the wings of the airplane. Tostudy the horizontal motion of a jet engine, it is modeled as a rigid body supportedby an elastic beam. The mass of the engine is m and the moment of inertia about itscentroidal axis C is J . The elastic beam is further modeled as a massless bar hingedat A, with the rotational spring κ providing restoring moment equal to κθ , whereθ is the angle between the bar and the vertical line as shown in Figure 5.25.

For small rotations, i.e.,∣∣θ ∣∣�1, set up the equation of motion for the jet engine

in term of θ . Find the natural frequency of oscillation.

AA

mg

m, J

RAx

RAy

κθκ

C

θ

JAθL

Figure 5.25 Horizontal vibration of a jet engine.

The system rotates about hinge A. The moment of inertia of the jet engine about itscentroidal axis C is J . Using the Parallel Axis Theorem, the moment of inertia ofthe jet engine about axis A is

JA = J + mL2.

Draw the free-body diagram of the jet engine and the supporting bar as shown.The jet engine is subjected to gravity mg . Remove the hinge at A and replace itby two reaction force components RAx and RAy . Since the bar rotates an angle θcounterclockwise, the rotational spring provides a clockwise restoring moment κθ .

Since the angular acceleration of the system is θ counterclockwise, the inertia

moment is JA θ clockwise.

Applying D’Alembert’s Principle, the free-body as shown in Figure 5.25 is indynamic equilibrium. Hence,

�

∑MA = 0 : JA θ + κθ + mg ·L sin θ = 0.

226 5 applications of linear differential equations

Example 5.7 — Single Degree-of-Freedom System 5.7

The single degree-of-freedom system described by x(t), as shown in Figure 5.26(a),is subjected to a sinusoidal load F(t)= F0 sin�t. Assume that the mass m, thespring stiffnesses k1 and k2, the damping coefficient c, and F0 and � are known.Determine the steady-state amplitude of the response of xP(t).

m

c

c

Ak1

k2

k2

F0 sin�t

k1

(a)

(b)

yk1y

k2(y−x) k2(x−y)cy

x(t)

A

mF0 sin�t

x, x, x

Figure 5.26 A vibrating system.

Introduce a displacement y(t) at A as shown in Figure 5.26(b). Consider the free-body of A . The extension of spring k1 is y and the compression of spring k2 isy−x. Body A is subjected to three forces: spring force k1 y, damping force c y, and

spring force k2(y−x). Newton’s Second Law requires

→ mA y = ∑F : mA y = −k1 y − c y − k2(y−x).

Since the mass of A is zero, i.e., mA = 0, one has

x = (k1 +k2)y + c y

k2. (1)

Consider the free-body of mass m. The extension of spring k2 is x−y. The massis subjected to two forces: spring force k2(x−y) and the externally applied loadF0 sin�t. Applying Newton’s Second Law gives

→ mx = ∑F : mx = F0 sin�t − k2(x−y).

Substituting equation (1) yields the equation of motion

m(k1 +k2) y + c

...y

k2= F0 sin�t − k2

[(k1 +k2)y + c y

k2− y

],

266 6 the laplace transform and its applications

Example 6.18 6.18

Solve y ′′′−y ′′+4 y ′−4 y = 40(t2 +t +1)H(t −2), y(0)= 5, y ′(0)= 0, y ′′(0)= 10.

Let Y(s)=L {y(t)

}. Taking the Laplace transform of both sides of the differential

equation yields

[s3 Y(s)− s2 y(0)− s y ′(0)− y ′′(0)

] − [s2 Y(s)− s y(0)− y ′(0)

]+ 4

[sY(s)− y(0)

] − 4Y(s) = L {40(t2 +t +1)H(t −2)

},

where, using L {f(t −a)H(t −a)

}= e−asL {f(t)

},

L {40(t2 +t +1)H(t −2)

} = 40L {[(t2 −4t +4)+5t −3

]H(t −2)

}= 40L {[

(t −2)2 +5(t −2)+7]

H(t −2)}

= 40e−2sL {t2 +5t +7

} = 40e−2s(2!

s3 + 5 · 1!s2 + 7 · 1

s

)L {

tn}= n!sn+1

= e−2s 40(7s2 +5s+2)

s3 .

Solving for Y(s) gives

Y(s) = 5s2 −5s+30

s3 −s2 +4s−4+ e−2s 40(7s2 +5s+2)

s3(s3 −s2 +4s−4).

Using partial fractions, one has

5s2 −5s+30

(s−1)(s2 +4)= A

s−1+ Bs+C

s2 +4= (A+B)s2 +(−B+C)s+(4A−C)

(s−1)(s2 +4)

To find A, cover-up (s−1) and set s = 1

A = 5s2 −5s+30

(s2 +4)

∣∣∣∣s=1

= 5−5+30

1+4= 6.

Comparing the coefficients of the numerators leads to

s2 : A + B = 5 =⇒ B = 5 − A = 5 − 6 = −1,

s : −B + C = −5 =⇒ C = B − 5 = −1 − 5 = −6,

1 : 4A − C = 30. Use this equation as a check: 4 ·6−(−6)= 30.

Hence,

L −1{

5s2 −5s+30

(s−1)(s2 +4)

}= L −1

{6

s−1+ −s−6

s2 +4

}

= L −1{

6 · 1

s−1− s

s2 +22 − 3 · 2

s2 +22

}= 6et − cos 2t − 3 sin 2t.

6.6 applications of the laplace transform 281

Example 6.23 — Beam-Column 6.23

Consider the beam-column shown in the following figure. Determine the lateraldeflection y(x).

a

EI, Lw

b

y

x

W

P P

Using the Heaviside step function and the Dirac delta function, the lateral load canbe expresses as

w(x) = w[1−H(x−a)

] + W δ(x−b).

Following the formulation in Section 5.4, the differential equation becomes

d4ydx4 + α2 d2y

dx2 = w[1−H(x−a)

] + W δ(x−b), α2 = PEI

, w = wEI

, W = WEI

.

Since the left end is a hinge support and the right end is a sliding support, theboundary conditions are

at x = 0 : deflection = 0 =⇒ y(0) = 0,

bending moment = 0 =⇒ y ′′(0) = 0,

at x = L : slope = 0 =⇒ y ′(L) = 0,

shear force = 0 =⇒ V(L) = −EI y ′′′(L)−P y ′(L) = 0

=⇒ y ′′′(L) = 0.

Applying the Laplace transform Y(s)=L{

y(x)}

, one has

[s4 Y(s) − s3 y(0) − s2 y ′(0) − s y ′′(0) − y ′′′(0)

] + α2 [s2 Y(s) − s y(0) − y ′(0)]

= ws

(1−e−as) + W e−bs.

Since y(0)= y ′′(0)= 0, solving for Y(s) leads to

Y(s) = y ′(0)

s2 + α2 +[

y ′′′(0)+α2y ′(0)]+W e−bs

s2(s2 +α2)+ w

s3(s2 +α2)(1−e−as).

Applying partial fractions

1s3(s2 +α2)

= As3 + B

s2 + Cs

+ D s+Es2 +α2 .

7.4 the matrix method 339

The general solution is

x(t) = X(t){

C +∫

X−1(t) f(t)dt}

=[

cos t sin t

sin t + cos t sin t − cos t

]{C1 +t + ln

∣∣cos t∣∣

C2 +t − ln∣∣cos t

∣∣}

,

∴ x1(t) = (t +C1) cos t + (t +C2) sin t + (cos t − sin t) ln∣∣cos t

∣∣,x2(t) = (C1−C2) cos t + (2t +C1 +C2) sin t + 2 cos t ln

∣∣cos t∣∣.

Example 7.20 7.20

Solve

x′(t)= A x(t)+f(t), x(t)=⎧⎨⎩

x1x2x3

⎫⎬⎭, A =

⎡⎣ 2 −1 −1

2 −1 −2−1 1 2

⎤⎦, f(t)=

⎧⎨⎩

2et

4e−t

0

⎫⎬⎭.

The characteristic equation is

det(A−λI) =∣∣∣∣∣∣2−λ −1 −1

2 −1−λ −2−1 1 2−λ

∣∣∣∣∣∣ = −(λ3 −3λ2 +3λ−1) = −(λ−1)3 = 0.

Hence, λ= 1 is an eigenvector of multiplicity 3. The eigenvector equation is

(A−λI)v =⎡⎢⎣

1 −1 −1

2 −2 −2

−1 1 1

⎤⎥⎦⎧⎪⎨⎪⎩

v1

v2

v3

⎫⎪⎬⎪⎭ =

⎧⎪⎨⎪⎩

v1 −v2 −v3

2(v1 −v2 −v3)

−(v1 −v2 −v3)

⎫⎪⎬⎪⎭ =

⎧⎪⎨⎪⎩

0

0

0

⎫⎪⎬⎪⎭,

which leads to v1 = v2 +v3. As a result, there are two linearly independent eigen-vectors. Taking v21 = 1 and v31 = −1, then v11 = v21 +v31 = 0,

∴ v1 =

⎧⎪⎨⎪⎩

v11

v21

v31

⎫⎪⎬⎪⎭ =

⎧⎪⎨⎪⎩

0

1

−1

⎫⎪⎬⎪⎭.

However, v2 cannot be chosen arbitrarily; it has to satisfy a condition imposed byv3, which will be clear in a moment.

A third linearly independent eigenvector does not exist. Hence, matrix A is de-fective and a complete basis of eigenvectors is obtained by including one generalizedeigenvector:

(A−λI)v3 = v2 =⇒

⎡⎢⎣

1 −1 −1

2 −2 −2

−1 1 1

⎤⎥⎦⎧⎪⎨⎪⎩

v13

v23

v33

⎫⎪⎬⎪⎭=

⎧⎪⎨⎪⎩

v13 −v23 −v33

2(v13 −v23 −v33)

−(v13 −v23 −v33)

⎫⎪⎬⎪⎭=

⎧⎪⎨⎪⎩

v12

v22

v32

⎫⎪⎬⎪⎭.

366 8 applications of systems of linear differential equations

8.2 Vibration Absorbers or Tuned Mass Dampers

In engineering applications, many systems can be modeled as single degree-of-freedom systems. For example, a machine mounted on a structure can be modeledusing a mass-spring-damper system, in which the machine is considered to be rigidwith mass m and the supporting structure is equivalent to a spring k and a damperc, as shown in Figure 8.2. The machine is subjected to a sinusoidal force F0 sin�t,which can be an externally applied load or due to imbalance in the machine.

x(t)

Supporting

Structure

F0 sin�t

c

m

k

Machine

Supporting Structure

MathematicalModeling

Figure 8.2 A machine mounted on a structure.

From Chapter 5 on the response of a single degree-of-freedom system, it is wellknown that when the excitation frequency � is close to the natural frequency ofthe system ω0 =√

k/m, vibration of large amplitude occurs. In particular, whenthe system is undamped, i.e., c = 0, resonance occurs when �=ω0, in which theamplitude of the response grows linearly with time.

To reduce the vibration of the system, a vibration absorber or a tuned massdamper (TMD), which is an auxiliary mass-spring-damper system, is mountedon the main system as shown in Figure 8.3(a). The mass, spring stiffness, anddamping coefficient of the viscous damper are ma, ka, and ca, respectively, wherethe subscript “a” stands for “auxiliary.”

To derive the equation of motion of the main mass m, consider its free-bodydiagram as shown in Figure 8.3(b). Since mass m moves upward, spring k isextended and spring ka is compressed.

❧ Because of the displacement x of mass m, the extension of spring k is x. Hencethe spring k exerts a downward force kx and the damper c exerts a downwardforce cx on mass m.

❧ Because the mass ma also moves upward a distance xa, the net compression inspring ka is x−xa. Hence the spring ka and damper ca exert downward forceska(x−xa) and ca(x− xa), respectively, on mass m.

Newton’s Second Law requires

↑ mx = ∑F : mx = −kx − cx − ka(x−xa)− ca(x− xa)+ F0 sin�t,

8.2 vibration absorbers or tuned mass dampers 367

x(t)

(a) (b)

VibrationAbsorber

xa(t)

ka(xa−x)

ka(x−xa)

kx cx

F0 sin�t

c

m

k

ca

ma

ka x(t)

xa(t)

F0 sin�t

c

m

k

ca

ma

ka

ca(xa−x)

ca(x−xa)

Figure 8.3 A vibration absorber mounted on the main system.

ormx + (c +ca) x + (k+ka)x − caxa − kaxa = F0 sin�t.

Similarly, consider the free-body diagram of mass ma. Since mass ma movesupward a distance xa(t), spring ka is extended. The net extension of spring ka isxa −x. Hence, the spring ka and damper ca exert downward forces ka(xa −x) andca(xa − x), respectively. Applying Newton’s Second Law gives

↑ ma xa = ∑F : ma xa = −ka(xa −x)− ca(xa − x),

∴ maxa + caxa + kaxa − cax − kax = 0.

The equations of motion can be written using the D-operator as[mD2 + (c +ca)D + (k+ka)

]x − (ca D +ka)xa = F0 sin�t,

−(ca D +ka)x + (ma D2 +ca D +ka)xa = 0.

Because of the existence of damping, the responses of free vibration (com-plementary solutions) decay exponentially and approach zero as time increases.Hence, it is practically more important and useful to study responses of forcedvibration (particular solutions). The determinant of the coefficient matrix is

φ(D) =∣∣∣∣∣mD2 + (c +ca)D + (k+ka) −(ca D +ka)

−(ca D +ka) ma D2 +ca D +ka

∣∣∣∣∣= [

mD2 + (c +ca)D + (k+ka)](ma D2 +ca D +ka)− (ca D +ka)

2

= [(mD2 +k)(ma D2 +ka)+ kama D2 + cac D2]

+ [ca(mD2 +k)+ c(ma D2 +ka)+ cama D2]D,

406 9 series solutions of differential equations

Example 9.8 9.8

Obtain series solution about x = 0 of the equation

2x2 y ′′ + x (2x + 1)y ′ − y = 0.

The differential equation is of the form

y ′′ + P(x)y ′ + Q(x)y = 0, P(x)= 2x+1

2x, Q(x)= − 1

2x2 .

Obviously, x = 0 is a singular point. Note that

x P(x) = 2x+1

2= 1

2 + x + 0 ·x2 + 0 ·x3 + · · · =⇒ P0 = 12 ,

x2Q(x) = − 12 = − 1

2 + 0 ·x + 0 ·x2 + 0 ·x3 + · · · =⇒ Q0 = − 12 .

Both x P(x) and x2Q(x) are analytic at x = 0 and can be expanded as power seriesthat are convergent for |x|<∞. Hence, x = 0 is a regular singular point.

The indicial equation is α(α−1)+αP0 +Q0 = 0:

α(α−1)+ α · 12 − 1

2 = 0 =⇒ (α+ 12 )(α−1) = 0 =⇒ α1 = 1, α2 = − 1

2 .

Thus the equation has a Frobenius series solution of the form

y1(x) = xα1

∞∑n=0

an xn =∞∑

n=0an xn+1, a0 �= 0, 0<x<∞,

where an, n = 0, 1, . . . , are constants to be determined. Differentiating with respectto x yields

y′1(x) =

∞∑n=0

(n+1)an xn, y′′1(x) =

∞∑n=1

(n+1)nan xn−1.

Substituting y1, y′1, and y′′

1 into the differential equation results in

∞∑n=1

2(n+1)nan xn+1 +∞∑

n=02(n+1)an xn+2 +

∞∑n=0

(n+1)an xn+1 −∞∑

n=0an xn+1 = 0.

Changing the indices of the summations

∞∑n=1

2(n+1)nan xn+1 n+1 = m==⇒∞∑

m=22m(m−1)am−1 xm,

∞∑n=0

2(n+1)an xn+2 n+2 = m==⇒∞∑

m=22(m−1)am−2 xm,

∞∑n=0

nan xn+1 n+1 = m==⇒∞∑

m=1(m−1)am−1 xm,

9.3 series solution about a regular singular point 407

one obtains

∞∑n=2

[2n(n−1)an−1 + 2(n−1)an−2

]xn +

∞∑n=1

(n−1)an−1 xn = 0.

For this equation to be true, the coefficient of xn, n = 1, 2, . . . , must be zero. Forn = 1, one has

0 ·a0 = 0 =⇒ a0 �= 0 is arbitrary; take a0 = 1.

For n � 2, one has

2n(n−1)an−1 + 2(n−1)an−2 + (n−1)an−1 = 0 =⇒ an−1 = − 2an−2

2n+1.

Hence,

n = 2 : a1 = − 2a0

2 ·2+1= − 2

5,

n = 3 : a2 = − 2a1

2 ·3+1= (−1)2 22

7 ·5,

...

n+1 : an = − 2an−1

2(n+1)+1= (−1)n 2n

(2n+3)(2n+1) · · · 5= (−1)n 3 ·2n

(2n+3)!! ,

where (2n+3)!!= (2n+3)(2n+1) · · · 5 ·3 ·1 is the double factorial. The firstFrobenius series solution is

y1(x) =∞∑

n=0an xn+1 =

∞∑n=0

(−1)n 3 ·2n

(2n+3)!! xn+1, 0<x<∞.

Since α1−α2 = 32 , according to Fuchs’ Theorem, a second linearly independent

solution is also a Frobenius series given by

y2(x) = xα2

∞∑n=0

bn xn =∞∑

n=0bn xn− 1

2 , b0 �= 0, 0<x<∞,

y′2(x) =

∞∑n=0

(n− 1

2)

bn xn− 32 , y′′

2(x) =∞∑

n=0

(n− 1

2)(

n− 32)

bn xn− 52 .

Substituting y2, y′2, and y′′

2 into the differential equation leads to

2x2∞∑

n=0

(n− 1

2)(

n− 32)

bn xn− 52 + (2x2 +x)

∞∑n=0

(n− 1

2)

bn xn− 32 −

∞∑n=0

bn xn− 12 = 0,

∞∑n=0

{[2(n− 1

2)(

n− 32) + (

n− 12) − 1

]bn xn− 1

2 + 2(n− 1

2)

bn xn+ 12

}= 0.

10.1 numerical solutions of first-order initial value problems 439

A schematic diagram is shown in Figure 10.3 to illustrate the procedure of theimproved Euler predictor-corrector method.

Improved Euler Predictor-Corrector Method

At the (i+1)th step, i = 0, 1, 2, . . . ,

(1) k1 = f(xi, yi) Slope at the left end point xi

(2) Predictor yPi+1 = yi + hk1 Predict y at xi+1 using the Euler method

(3) k2 = f(xi+1, yPi+1) Predicted slope at the right end point xi+1

(4) k = k1 + k2

2The averaged slope is used on [xi, xi+1]

(5) Corrector yi+1 = yi + hk Improved Euler point

y

xi+1 xi+2xi

yi

Slope k1= f(xi , yi)

Average slope k=

x

Exact valuey(xi+1)

2

3

4

1

Improved Euler point yi+1=yi +hk

Slope k2= f(xi+1, yi+1)

k1+k22

5

Exact solution

h h

Euler point yi+1=yi +hk1P

P

Figure 10.3 Improved Euler predictor-corrector method.

Example 10.3 10.3

For the initial value problem y ′ = x y2 −y, y(0)= 0.5, determine y(1.0) using theimproved Euler method and the improved Euler predictor-corrector method withh = 0.5.

(1) The improved Euler method is, with f(x, y)= x y2 − y and h = 0.5,

yi+1 = yi + 12 h

[f(xi, yi)+ f(xi+1, yi+1)

].

The results are as follows

i = 0 : x0 = 0, y0 = 0.5,

480 11 partial differential equations

11.4.3 One-Dimensional Transient Heat Conduction

Consider a wall or plate of infinite size and of thickness L, as shown in Figure11.7, which is suddenly exposed to fluids in motion on both of its surfaces. Thecoefficient of thermal conductivity of the wall or plate is k. Suppose the wall has aninitial temperature distribution T(x, 0)= f(x). The temperatures of the fluids andthe heat transfer coefficients on the left-hand and right-hand sides of the wall areTf 1, h1 and Tf 2, h2, respectively.

∞

∞

h1, Tf1 h2, Tf2

T(x,0)= f(x)

x=Lx=0 x

L

k, α

Figure 11.7 An infinite wall.

Because the wall or plate is infinitely large, the heat transfer process is simplifiedas one-dimensional (in the x-dimension).

The differential equation (Fourier’s equation in one-dimension), the initial con-dition, and the boundary conditions of this one-dimensional transient heat con-duction problem are

∂T

∂t= α

∂2T

∂x2 , 0 � x � L, t � 0,

Initial Condition (IC) : T = f(x), at t = 0,

Boundary Conditions (BCs) : k∂T

∂x= h1 (T −Tf 1), at x = 0,

−k∂T

∂x= h2 (T −Tf 2), at x = L.

This mathematical model has many engineering applications.

❧ The infinite wall is a model of a flat wall of a heat exchanger, which is initiallyisothermal at T = T0. The operation of the heat exchanger is initiated at t = 0;two different fluids of temperatures Tf 1 and Tf 2, respectively, are flowingalong the sides of the wall.

❧ The infinite wall is a model of a wall in a building or a furnace. One side of thewall is suddenly exposed to a higher temperature Tf 1 due to fire occurring ina room or the ignition of flames in the furnace.

12.3 numerical solutions of differential equations 519

Example 12.23 — Dynamical Response of Parametrically Excited System 12.23

Consider the parametrically excited nonlinear system given by

x + β x − (1+μ cos�t)x + αx3 = 0.

Examples of this equation are found in many applications of mechanics, especiallyin problems of dynamic stability of elastic systems. In particular, the transversevibration of a buckled column under the excitation of a periodic end displacementis described by this equation. The system is called parametrically excited becausethe forcing term μ cos�t appears in the coefficient (parameter) of the equation.

It is a good practice to put restart at the beginning of each program so thatMaple can start fresh if the program has to be rerun.>restart:

>with(plots): Load the plots package.

>ODE:=diff(x(t),t$2)+beta*diff(x(t),t)-(1+mu*cos(Omega*t))*x(t)

+alpha*x(t)^3=0: Define the ODE.

>ICs:=x(0)=0,D(x)(0)=0.1: Define the ICs: x(0)= 0, x(0)= 0.1.

Periodic Motion (μ= 0.3)>alpha:=1.0: beta:=0.2: Omega:=1.0: mu:=0.3: Assign the parameters.

Solve the system numerically using dsolvewith option numeric.>sol:=dsolve({ODE,ICs},x(t),numeric,maxfun=1000000):

Plot the time series x(t) versus t , (x1 = x). Figure 12.1(a)>odeplot(sol,[t,x(t)],t=0..500,numpoints=10000,labels=["t","x1"],

tickmarks=[[0,100,200,300,400,500],[-1.5,-1,-0.5,0,0.5,1,1.5]]);

Plot the time series x(t) versus t , (x2 = x). Figure 12.1(b)>odeplot(sol,[t,D(x)(t)],t=0..500,numpoints=10000,labels=["t","x2"],

tickmarks=[[0,100,200,300,400,500],[-0.8,-0.6,-0.4,-0.2,0,0.2,0.4,

0.6,0.8]]);

Plot the phase portrait x(t) versus x(t). Figure 12.1(c)>odeplot(sol,[x(t),D(x)(t)],t=0..500,numpoints=10000,view=[-1.8..1.8,

-1.0..1.0], tickmarks=[[-1.8,-1.2,-0.6,0.6,1.2,1.8],[-1,-0.75,-0.5,

-0.25,0.25,0.5,0.75,1]],axes=normal,labels=["x1","x2"]);

When μ= 0.3, after some transient part, the response of the systemwill settledown to periodic motion.

Chaotic Motion (μ= 0.4)>alpha:=1.0: beta:=0.2: Omega:=1.0: mu:=0.4: Assign the parameters.

12.3 numerical solutions of differential equations 521

–1.5

–1

–0.5

0

0.5

1

1.5

100 200 300 400 500

t

x1

(a)

–0.8

–0.6

–0.4

–0.2

00.2

0.4

0.6

0.8

100 200 300 400 500

t

x2

(b)

–1

–0.75

–0.5

–0.25

0.25

0.5

0.75

1

–1.8 –1.2 –0.6 0.6 1.2 1.80

x2

x1

(c)

Figure 12.2 Chaotic motion.

Plot the time series x(t) versus t . Figure 12.2(b)>odeplot(sol,[t,D(x)(t)],t=0..500,numpoints=10000,labels=["t","x2"],

tickmarks=[[0,100,200,300,400,500],[-0.8,-0.6,-0.4,-0.2,0,0.2,0.4,

0.6,0.8]]);

Plot the phase portrait x(t) versus x(t). Figure 12.2(c)>odeplot(sol,[x(t),D(x)(t)],t=0..500,numpoints=10000,view=[-1.8..1.8,

-1.0..1.0], tickmarks=[[-1.8,-1.2,-0.6,0.6,1.2,1.8],[-1,-0.75,-0.5,

-0.25,0.25,0.5,0.75,1]],axes=normal,labels=["x1","x2"]);

Index

AAmplitude, 195, 201, 208

modulated, 208Analytic function, 393Ascending motion of a rocket problem, 421Automobile ignition circuit problem, 209Auxiliary equation, 144Average slope method, 437, 446, 453

BBackward Euler method, 436, 446, 449, 453Bar with variable cross-section problem, 121Base excitation, 190–191Beam-column, 218

boundary conditions, 219equation of equilibrium, 220Laplace transform, 280

Beamflexural motion, 465

equation of motion, 465forced vibration, 471free vibration, 466

Beams on elastic foundation, 283boundary conditions, 284equation of equilibrium, 284problem, 284, 288

Beat, 208Bernoulli differential equation, 58, 75Bessel’s differential equation, 390, 408, 420,

424–426application, 418series solution, 408, 418solution, 418

Bessel function, 418, 425Maple, 512of the first kind, 410–411, 415, 418, 425of the second kind, 412–413, 415, 418, 425

Body cooling in air problem, 87Boundary value problem, 10Buckling of a tapered column problem, 418

Maple, 513Buckling

column, 221

tapered column, 418Maple, 513

Bullet through a plate problem, 94

CCable of a suspension bridge problem, 100Cable under self-weight, 102Capacitance, 108Capacitor, 108Chain moving problem, 123, 125Characteristic equation, 143–144, 147, 151, 180,

305, 344, 460complex roots, 147linear differential equation, 144, 180multiple degrees-of-freedom system, 344real distinct roots, 143–144repeated roots, 151

Characteristic number, 144, 460Circuit, 108, 209, 275, 278, 372–373, 375

first-order, 113parallel RC, 110parallel RL, 111parallel RLC, 209RC, 109RL, 109second-order, 213

Laplace transform, 275, 278series RC, 110series RL, 110series RLC, 209system of linear differential equations

Laplace transform, 373matrix method, 375

Clairaut equation, 67Complementary solution

linear differential equation, 142–144, 147, 152,180, 460

matrix method, 326, 350complex eigenvalues, 328–329, 349distinct eigenvalues, 326–327, 349multiple eigenvalues, 330–331, 349

method of operator

542

index 543

system of linear differential equations,304–305, 348

Complex numbers, 148Constant slope method, 432–433, 445Convolution integral, 258Cooling, 87Cover-up method, 260Cramer’s Rule, 307Critically damped system, 199

Laplace transform, 279Cumulative error, 435, 452–453

DD’Alembert’s Principle, 91, 223, 229, 358–359,

364–365D-operator, 140, 162

inverse, 162properties, 141

Damped natural circular frequency, 194, 196Damped natural frequency, 196Damper, 189, 302Damping, 189, 302Damping coefficient, 193–194, 198–199

modal, 347Damping force, 189–190, 302Dashpot damper, 189, 302Defective (deficient) matrix, 331Degree-of-freedom system

four, 362infinitely many, 468multiple, 301, 303, 344

damped forced vibration, 346orthogonality of mode shapes, 345undamped forced vibration, 346undamped free vibration, 344

single, 188, 191Laplace transform, 268response, 193

two, 357, 377Derivatives table, 533Differential equation

boundary value problem, 10definition, 6existence and uniqueness, 11–12general solution, 8homogeneous, 7, 20initial value problem, 10

numerical solution, 431linear, 6–7, 140

constant coefficients, 7, 140variable coefficients, 7, 178, 183

Maple, 498–499Mathematica, 498nonhomogeneous, 7nonlinear, 6numerical solution, 431

average slope method, 437, 453backward Euler method, 436, 449, 453constant slope method, 432–433cumulative error, 435, 452–453error analysis, 434Euler method, 432–433, 449, 452explicit method, 452forward Euler method, 432–433, 452GNU Scientific Library, 453implicit method, 437–438, 453improved Euler method, 437, 449, 453improved Euler predictor-corrector method,

439, 452IMSL Library, 453local error, 434–435, 453Maple, 453Mathematica, 453Matlab, 453midpoint method, 441NAG Library, 453Numerical Recipes, 453predictor-corrector technique, 438roundoff error, 434Runge-Kutta-Fehlberg method, 517–518Runge-Kutta method, 440–444, 452trapezoidal rule method, 438truncation error, 434–435, 453

order, 6ordinary, 6partial, 8particular solution, 8series solution, 390

Frobenius series, 403, 405, 425Fuchs’ Theorem, 405indicial equation, 404, 425Maple, 512ordinary point, 394, 397regular singular point, 403

singular solution, 19stiff, 449–450, 453

Maple, 517Dirac delta function, 254–256

concentrated load, 471Laplace transform, 256Maple, 507properties, 255

544 index

Direction field, 11Displacement meter problem, 229Dynamic magnification factor (DMF), 202, 370Dynamical response of parametrically excited

system problem, 518

EEarthquake, 190Eigenvalue, 326, 328–331, 349

Maple, 510Eigenvector, 326, 328–331, 349

generalized, 331, 349Maple, 510

Error analysis, 434Euler’s formula, 147, 149Euler constant, 413Euler differential equation, 178, 183Euler method, 432–433, 445, 449, 452Exact differential equation, 31–33, 76Example

ascending motion of a rocket, 421automobile ignition circuit, 209bar with variable cross-section, 121beam-column

Laplace transform, 280beams on elastic foundation, 284, 288body cooling in air, 87buckling of a tapered column, 418

Maple, 513bullet through a plate, 94cable of a suspension bridge, 100chain moving, 123, 125displacement meter, 229dynamical response of parametrically excited

system, 518ferry boat, 120float and cable, 107flywheel vibration, 227free flexural vibration of a simply supported

beam, 468heating in a building, 88jet engine vibration, 223Lorenz system, 522object falling in air, 95particular moving in a plane, 300piston vibration, 224reservoir pollution, 127second-order circuit

Laplace transform, 275single degree-of-freedom system under blast

force, 273

single degree-of-freedom system undersinusoidal excitation, 270

two degrees-of-freedom system, 357vehicle passing a speed bump, 213

Laplace transform, 272vibration of an automobile, 362water leaking, 126water tower, 220

Excitation frequency, 202, 204, 208Existence and uniqueness theorem, 12Explicit method, 452Externally applied force, 189, 191

FFerry boat problem, 120Finned surface, 476–477, 488First-order circuit, 113

problem, 111–112First-order differential equation

Bernoulli, 58, 75Clairaut, 67exact, 31–33, 76homogeneous, 20, 75inspection, 45, 76integrating factor, 31, 39–40, 76integrating factor by groups, 48, 77linear, 55, 75Maple, 499separation of variables, 16, 20, 75solvable for dependent variable, 61, 77solvable for independent variable, 61–62, 77special transformation, 25, 77

Flexural motion of beam, 465equation of motion, 465forced vibration, 471

separation of variables, 471free vibration, 466

infinitely many degrees-of-freedom system,468

separation of variables, 466Float and cable problem, 107Flywheel vibration problem, 227Forced vibration, 193

multiple degrees-of-freedom system, 346single degree-of-freedom system, 200

Laplace transform, 270, 278two-story shear building, 380

Forward Euler method, 432–433, 445, 452Fourier’s equation in one-dimension, 474, 480,

483Fourier’s Law of Heat Conduction, 473

index 545

Fourier integral, 485cosine integral, 485sine integral, 485–486

Fourier series, 470, 485cosine series, 471sine series, 469, 471, 479, 482, 491–492

Free flexural vibration of a simply supportedbeam problem, 468

Free vibration, 193multiple degrees-of-freedom system, 344single degree-of-freedom system

critically damped, 199Laplace transform, 269, 278overdamped, 199undamped, 194–195underdamped, 196

two-story shear building, 378Frequency equation, 344, 378

multiple degrees-of-freedom system, 344Frobenius series, 403, 405, 425Fuchs’ Theorem, 405Fundamental Theorem, 141

GGamma function, 410

Maple, 512Gauss-Jordan method, 335General solution, 8, 380

linear differential equation, 142matrix method, 335, 350method of operator, 312, 348

Generalized eigenvector, 331, 349GNU Scientific Library, 453Grouping terms method, 34

HHanging cable, 97

cable under self-weight, 102problem, 106suspension bridge, 97

Heat conduction, 473–474, 476boundary conditions, 475

homogeneous, 476insulated, 475of the first kind, 475of the second kind, 475of the third kind, 475

coefficient of thermal conductivity, 473equation, 473

one-dimensional transient, 474three-dimensional steady-state, 475

two-dimensional steady-state, 474finned surface, 476–477, 488Fourier’s equation in one-dimension, 474, 480,

483Fourier’s Law, 473heat transfer coefficient, 476Laplace’s equation

in three-dimensions, 475, 488in two-dimensions, 474, 477

one-dimensional transient, 480on a semi-infinite interval, 483

thermal diffusivity, 474three-dimensional steady-state, 488two-dimensional steady-state, 476

Heating, 87Heating in a building problem, 88Heaviside step function, 249–250, 255

inverse Laplace transform, 257Laplace transform, 249, 252Maple, 507

Higher-order differential equationdependent variable absent, 70, 77immediately integrable, 68, 77independent variable absent, 72, 78Maple, 499

Homogeneousdifferential equation, 7, 20first-order differential equation, 20, 75

Hyperbolic functions, 145

IImplicit method, 438, 453Improved Euler method, 437, 446, 449, 453Improved Euler predictor-corrector method,

439, 446, 452Impulse-Momentum Principle, 91, 125, 254, 422Impulse function, 254–255IMSL Library, 453Indicial equation, 404, 425Inductance, 109Inductor, 109Inertia force, 91, 230–231, 359, 364–365Inertia moment, 223, 229, 359, 364Infinitely many degrees-of-freedom system, 468Initial value problem, 10

numerical solution, 431Inspection method, 45Integral transform, 244Integrals table, 534Integrating factor, 31, 39–40, 76Integrating factor by groups, 48, 77

546 index

Inverse Laplace transform, 257convolution integral, 258Heaviside step function, 257Maple, 507properties, 257, 539table, 257, 539–540

JJet engine, 192Jet engine vibration problem, 223

KKirchhoff’s Current Law (KCL), 108, 110–113, 209,

212–213, 277, 373Kirchhoff’s Voltage Law (KVL), 108, 110, 209, 211,

276–277, 373

LLanding gear, 192Laplace’s equation

in three-dimensions, 475, 488in two-dimensions, 474, 477

Laplace transform, 244beam-column, 280beams on elastic foundation, 283convolution integral, 258definition, 244Dirac delta function, 256Heaviside step function, 249, 252inverse, 257

properties, 257, 539table, 539

linear differential equation, 263Maple, 507properties, 245, 537second-order circuit, 275single degree-of-freedom system, 268, 278

blast force, 273forced vibration, 270, 278free vibration, 269, 278

system of linear differential equations, 318, 348table, 245, 537–538vehicle passing a speed bump problem, 272

Legendre equation, 397Linear differential equation, 140

auxiliary equation, 144characteristic equation, 143–144, 147, 151, 180,

460complex roots, 147real distinct roots, 143–144repeated roots, 151

characteristic number, 144, 460complementary solution, 142–144, 147, 152, 180,

460constant coefficients, 7, 140general solution, 142Laplace transform, 263Maple, 506, 512particular solution, 142, 153, 181

method of operator, 162–164, 166, 169, 171,181–182

method of undetermined coefficients, 153,181,

method of variation of parameters, 173,181–182

variable coefficients, 7, 178, 183second-order homogeneous, 403

Linear first-order differential equation, 55, 75Lipschitz condition, 12Local error, 434–435, 453Lorenz system problem, 522

MMaple, 453, 498algsubs, 502–503assuming, 502, 504BesselJ, 512–513BesselY, 512–513collect, 502, 504, 506, 508–510, 512, 514–515,

517cos, 506, 508, 510exp, 506, 508–510Heaviside, 508–509ln, 504, 517sin, 506, 508, 510

combine, 502, 504, 508convert, 507, 512, 514–517factorial, 512, 517parfrac, 507polynom, 512, 514–517

cos, 499, 502, 509, 519D, 506(D@@n)(y)(a), 506D(y)(a), 506

diff, 499–504, 506, 508–509, 514–516, 518–519,522

Dirac, 507do loop, 525dsolve, 498–510, 514–520, 522, 525’formal_solution’, 515,–516implicit, 499–503, 505method=laplace, 507–508

index 547

numeric, 517–520, 522, 525,–520, 522,525,–518

series, 514,–517Eigenvectors, 510–511, 517eval, 505evalf, 512, 514, 518exp, 500, 506–509fsolve, 514GAMMA, 512gamma, 512Heaviside, 507–509I, 510implicitplot, 504labels, 504numpoints, 504thickness, 504tickmarks, 504view, 504

invlaplace, 507JordanForm, 511laplace, 507lhs, 501–503ln, 501, 506, 512–513map, 500Matrix, 510–511, 517numer, 501–503odeplot, 519–523axes, 519, 521, 523labels, 519–523numpoints, 519–523orientation, 523thickness, 523tickmarks, 519–523view, 519, 521, 523

odetest, 500, 504, 510Order, 515plot, 513numpoints, 513

pointplot, 525axesfont, 525color, 525labelfont, 525labels, 525style, 525tickmarks, 525

positive, 502, 504restart, 519, 522rhs, 505, 525RootOf, 505seq, 525series, 512, 514

simplify, 502, 505, 512, 517sqrt, 502

sin, 499, 502, 506–507, 509solve, 504–505sqrt, 501subs, 504–505, 512–513, 517unapply, 505, 518with, 504, 507, 510–511, 517, 519, 522inttrans, 507LinearAlgebra, 510–511, 517plots, 504, 519, 522

zip, 525Mass-spring-damper system, 191Mathematica, 453, 498Matlab, 453Matrix method, 325, 349

complementary solution, 326, 350complex eigenvalues, 328–329, 349distinct eigenvalues, 326–327, 349multiple eigenvalues, 330–331, 349

general solution, 335, 350particular solution, 334, 350

method of variation of parameters, 334system of linear differential equations, 325, 349

MatrixCramer’s Rule, 307damping, 303, 346–347defective (deficient), 331eigenvalue, 326, 328–331, 349eigenvector, 326, 328–331, 349Gauss-Jordan method, 335generalized eigenvector, 331, 349inverse, 335–336

Gauss-Jordan method, 335Maple, 510mass, 303, 344–346modal, 345, 380stiffness, 303, 344–346

Mechanical vibration, 357Method of grouping terms, 34Method of inspection, 45, 76Method of operator

linear differential equations, 162, 181polynomial, 166, 181Shift Theorem, 164, 182system of linear differential equations,

304–305, 307–308, 347–348Theorem 1, 163, 182Theorem 2, 164, 182Theorem 3, 169, 182Theorem 4, 171, 182

548 index

Method of separation of variablesfirst-order differential equation, 16, 20, 75partial differential equation, 458, 492

Method of undetermined coefficients, 153, 181exception, 159

Method of variation of parameterslinear differential equations, 173, 181–182system of linear differential equations, 314, 334

Midpoint method, 441Modal damping coefficient, 347Modal frequency, 344, 378Modal matrix, 345, 380Mode shape, 344, 379–380, 468

orthogonality, 345, 380Moment-curvature relationship, 219, 419, 465Moment of inertia, 223, 228, 358, 362, 419, 465

Parallel Axis Theorem, 223, 228Motion, 91Multiple degrees-of-freedom system, 301, 344

damped forced vibration, 346equations of motion, 303orthogonality of mode shapes, 345undamped forced vibration, 346undamped free vibration, 344

NNAG Library, 453Natural circular frequency, 193, 195

damped, 194undamped, 193

Natural frequency, 195, 204, 208Natural purification in a stream, 114Newton’s Law of Cooling, 87Newton’s Second Law, 2, 91, 93, 95–96, 123, 190,

214, 224, 226, 301–302, 366–367, 465Numerical Recipes, 453Numerical solution, 431

average slope method, 437, 446, 453backward Euler method, 436, 446, 449, 453conditionally stable, 449constant slope method, 432–433, 445cumulative error, 435, 452–453error analysis, 434Euler method, 432–433, 445, 449, 452explicit method, 452forward Euler method, 432–433, 445, 452GNU Scientific Library, 453implicit method, 437–438, 453improved Euler method, 437, 446, 449, 453improved Euler predictor-corrector method,

439, 446, 452

IMSL Library, 453local error, 434–435, 453Maple, 453, 517Mathematica, 453Matlab, 453midpoint method, 441NAG Library, 453Numerical Recipes, 453predictor-corrector technique, 438roundoff error, 434Runge-Kutta-Fehlberg method, 517–518Runge-Kutta method, 440

fourth-order, 443–444, 446, 452second-order, 441–442, 446, 452

stability, 449, 451, 453stepsize, 431, 453system of differential equations, 445trapezoidal rule method, 438truncation error, 434–435, 453unconditionally stable, 449unstable, 449

OObject falling in air problem, 95Ohm’s Law, 108Operator

D, 140, 162inverse, 162properties, 141

method, 162, 304linear differential equations, 162, 181polynomial, 166, 181Shift Theorem, 164, 182system of linear differential equations,

304–305, 307–308, 347–348Theorem 1, 163, 182Theorem 2, 164, 182Theorem 3, 169, 182Theorem 4, 171, 182

Ordinary differential equation, 6Ordinary point, 396, 424Orthogonality

mode shape, 380sine and cosine functions, 470, 472, 482

Overdamped system, 199Laplace transform, 279

PParallel Axis Theorem, 223, 228Parallel circuit

RC, 110

index 549

RL, 111RLC, 209

Partial differential equation, 8, 457separation of variables, 458, 492simple, 457

Partial fractions, 259cover-up method, 260Maple, 507

Particular moving in a plane problem, 300polar coordinates, 301rectangular coordinates, 300

Particular solution, 8, 193, 200linear differential equation, 142, 153, 181matrix method, 334, 350

method of variation of parameters, 334method of operator

linear differential equations, 162, 181polynomial, 166, 181Shift Theorem, 164, 182system of linear differential equations,

307–308, 348Theorem 1, 163, 182Theorem 2, 164, 182Theorem 3, 169, 182Theorem 4, 171, 182

method of undetermined coefficients, 153, 181exception, 159

method of variation of parameterslinear differential equations, 173, 181–182system of linear differential equations, 314,

334Period, 195Phase angle, 195, 201Piston vibration problem, 224Power series, 391

convergence, 391Maple, 512operation, 392radius of convergence, 391

Predictor-corrector technique, 438

RRC circuit, 109Reservoir pollution problem, 127Resisting medium, 91Resonance, 203–204, 208, 382–383RL circuit, 109RLC circuit, 209Roundoff error, 434Runge-Kutta-Fehlberg method, 517–518Runge-Kutta method, 440

fourth-order, 443–444, 446, 452second-order, 441–442, 446, 452

SSecond-order circuit, 213

Laplace transform, 275, 278problem, 211, 275

Separation of variables methodfirst-order differential equation, 16, 20, 75partial differential equation, 458, 492

Series circuitRL, 110RLC, 209RC, 110

Series solution, 390Bessel’s differential equation, 408, 418Frobenius series, 403, 405, 425Fuchs’ Theorem, 405indicial equation, 404, 425Legendre equation, 397linear differential equation, 403Maple, 512, 514ordinary point, 394, 397regular singular point, 403

Shear buildingmultiple story, 301single story, 188–191two-story, 377

forced vibration, 380free vibration, 378

Shear force, 189–190, 302Single degree-of-freedom system, 188, 191, 193

blast forceLaplace transform, 273problem, 273

critically damped, 199Laplace transform, 279

Laplace transform, 268, 278blast force, 273forced vibration, 270, 278free vibration, 269, 278sinusoidal excitation, 270

overdamped, 199Laplace transform, 279

problem, 226response, 193sinusoidal excitation

Laplace transform, 270problem, 270

undamped, 194, 204, 208underdamped, 194, 200

550 index

Laplace transform, 278Singular point, 403

irregular, 403regular, 403, 425

Singular solution, 19Sinusoidal excitation, 200, 204Special function

Maple, 512Special transformation

first-order differential equation, 25, 77Steady-state solution, 200Stepsize, 431, 453Stiff differential equation, 449–450, 453

Maple, 517Stiffness, 189Stream, 114Suspension bridge, 97System of differential equations

Maple, 509numerical solution, 445

average slope method, 446backward Euler method, 446constant slope method, 445Euler method, 445forward Euler method, 445fourth-order Runge-Kutta method, 446improved Euler method, 446improved Euler predictor-corrector method,

446second-order Runge-Kutta method, 446

System of linear differential equationscomplementary solution

complex eigenvalues, 328–329, 349distinct eigenvalues, 326–327, 349matrix method, 326, 350method of operator, 304–305, 348multiple eigenvalues, 330–331, 349

general solutionmatrix method, 335, 350method of operator, 312, 348

Laplace transform, 318, 348matrix method, 325, 349method of operator, 304, 347

characteristic equation, 305particular solution

matrix method, 334, 350method of operator, 307–308, 348method of variation of parameters, 314, 334

TTable

derivatives, 533integrals, 534inverse Laplace transform, 539Laplace transform, 537trigonometric identities, 531

Taipei 101, 371–372Taylor series, 432, 440Transient solution, 200Trapezoidal rule method, 438Trigonometric identities table, 531Truncation error, 434–435, 453Tuned mass damper (TMD), 366–367, 370–372Two degrees-of-freedom system

problem, 357shear building, 377

forced vibration, 380free vibration, 378

UUndamped system, 194, 204, 208Underdamped system, 194, 200

Laplace transform, 278Undetermined coefficients method, 153, 181

VVariation of parameters method

linear differential equations, 173, 181–182system of linear differential equations, 314, 334

Vehicle passing a speed bump problem, 213Laplace transform, 272

Vibration, 188–191, 193, 213, 272, 301, 344, 357, 377,468

Vibration absorber, 366–367, 370–372Vibration of an automobile problem, 362Viscous dashpot damper, 189

WWater leaking problem, 126Water tower problem, 220